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arxiv: 2602.18994 · v2 · pith:UGOHGREInew · submitted 2026-02-22 · 🌀 gr-qc · astro-ph.GA

Testing the Spacetime Geometry of Sgr A* with the Relativistic Orbit of S2 star

Parth Bambhaniya , Preet Dalal , Giovani H. Vicentin , Riccardo Della Monica , Elisabete M. de Gouveia Dal Pino , Bina Patel This is my paper

Pith reviewed 2026-05-22 10:20 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GA
keywords Sgr A*S2 starrelativistic geodesicsblack hole spacetimesVLT astrometryEvent Horizon Telescopenaked singularitiesspacetime tests
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The pith

Observations of the S2 star cannot yet distinguish Schwarzschild, Reissner-Nordström, and Bardeen spacetimes around Sgr A*.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper integrates the timelike geodesic motion of the S2 star in a range of compact-object metrics and projects the paths onto observable astrometric positions and radial velocities, adding Rømer delay and relativistic redshift. These predictions are compared directly with existing VLT measurements of S2 while also requiring each metric to reproduce the Event Horizon Telescope shadow size. The result is that several geometries which produce identical shadows remain statistically equivalent when fitted to the current S2 data set. A sympathetic reader cares because this shows how much orbital precision is still needed before stellar tracks can break the degeneracies that shadow images leave intact.

Core claim

By solving the timelike geodesic equations in Schwarzschild, Reissner-Nordström, Bardeen, Hayward, Simpson-Visser, and Janis-Newman-Winicour geometries, projecting the resulting orbits with Rømer delay and redshift, and fitting to VLT astrometry and spectroscopy under the EHT shadow constraint, the analysis finds that Schwarzschild, Reissner-Nordström, and Bardeen models remain statistically indistinguishable with present S2 observations.

What carries the argument

Numerical integration of timelike geodesics followed by projection onto astrometric and spectroscopic observables that include Rømer time delay and gravitational redshift, performed for each metric and compared to VLT data plus EHT shadow size.

If this is right

  • Schwarzschild, Reissner-Nordström, and Bardeen geometries produce statistically equivalent fits to current S2 data.
  • Upper limits are obtained on the charge-like parameter q/M for non-Schwarzschild models.
  • Akaike and Bayesian information criteria rank the relative performance of the alternative metrics.
  • Higher-precision astrometry expected from upcoming VLT and Keck campaigns can break the remaining degeneracies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Stellar-orbit and shadow observations appear to constrain partially independent combinations of metric parameters, preserving degeneracies that each method alone cannot resolve.
  • Including perturbations from an extended mass distribution or nearby stars in the orbit model could reduce the number of viable geometries.
  • Applying the same geodesic-fitting procedure to additional stars with measured orbits around Sgr A* would increase the statistical power of the test.

Load-bearing premise

The observed positions and velocities of S2 are produced solely by geodesic motion in the chosen spacetime plus the modeled delay and redshift effects, with no significant extra contributions from other stars, gas, or instrumental systematics.

What would settle it

A future VLT or Keck measurement of S2's position or velocity at pericenter that deviates from the orbit predicted by one of the currently equivalent models while remaining consistent with another would demonstrate that the models are distinguishable.

Figures

Figures reproduced from arXiv: 2602.18994 by Bina Patel, Elisabete M. de Gouveia Dal Pino, Giovani H. Vicentin, Parth Bambhaniya, Preet Dalal, Riccardo della Monica.

Figure 1
Figure 1. Figure 1: FIG. 1: Constraints on the generalized charge-like parameter [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Figure shows the marginalized posterior [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: This figure shows the combined astrometric and spectroscopic orbital fits of the S2 star using 24 years of VLT [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: This figure shows the corner plot of the Schwarzschild model obtained from an MCMC analysis of the [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: This figure shows the corner plot of the RN model obtained from an MCMC analysis of the combined [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: This figure shows the corner plot of the BD model obtained from an MCMC analysis of the combined [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
read the original abstract

In this work, we perform a relativistic test of the spacetime geometry of Sagittarius A* (Sgr A*) using the orbit of the S2 star. We consider a broad class of compact object models, including Schwarzschild, Reissner-Nordstr\"om, Bardeen, Hayward, and Simpson-Visser black holes, as well as the Janis-Newman-Winicour naked singularity spacetime. For each geometry, we integrate the timelike geodesic equations and consistently project the resulting trajectories onto astrometric and spectroscopic observables, incorporating R{\o}mer time delay and relativistic redshift effects. The theoretical predictions are tested with current Very Large Telescope (VLT) observations of the S2 star, while simultaneously imposing constraints from the Event Horizon Telescope shadow size. We find that several spacetimes that are degenerate at the level of shadow imaging, most notably Schwarzschild, Reissner-Nordstr\"om, and Bardeen regular black hole geometries, remain statistically indistinguishable when tested against present S2 data. We further carry out a statistical model comparison based on the Akaike and Bayesian information criteria (AIC and BIC) to evaluate the relative performance of the alternative spacetime models. Our analysis also constrains the generalized charge like parameter $q/M$ in non-Schwarzschild spacetimes based on current S2 star observations, and identifies specific black hole and horizonless geometries that can be further tested with forthcoming high precision astrometric observations from the VLT and Keck telescopes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript integrates timelike geodesics in Schwarzschild, Reissner-Nordström, Bardeen, Hayward, Simpson-Visser, and Janis-Newman-Winicour spacetimes, projects the resulting S2 orbits to astrometric positions and radial velocities including Rømer delay and relativistic redshift, fits the models to VLT data while imposing the EHT shadow-size constraint, performs AIC/BIC model comparison, and derives bounds on the generalized charge parameter q/M. It concludes that several geometries degenerate under shadow imaging remain statistically indistinguishable with current S2 observations.

Significance. If the central result holds, the work establishes that present S2 astrometry and spectroscopy, even with full relativistic projection, do not yet resolve the shadow-level degeneracies among Schwarzschild, RN, and Bardeen geometries, thereby supplying a quantitative baseline for the precision gains expected from forthcoming VLT and Keck campaigns. The explicit AIC/BIC comparison and joint EHT+S2 analysis constitute a reproducible, falsifiable test of the models.

major comments (2)
  1. [§4 and abstract] §4 (model comparison) and the abstract: the claim that Schwarzschild, RN, and Bardeen remain statistically indistinguishable rests on the unquantified assumption that residuals in the VLT astrometric and radial-velocity data are dominated by measurement noise rather than by unmodeled extended mass or stellar perturbations. The manuscript must demonstrate that the reported ΔAIC/ΔBIC values remain stable when an extended-mass component (e.g., a power-law cusp) is added to the fit; otherwise the indistinguishability conclusion is not load-bearing.
  2. [§3.1, Eq. (12)] §3.1 (geodesic integration) and Eq. (12): the projection operator includes Rømer delay and gravitational redshift but does not marginalize over possible offsets between the coordinate origin and the radio center of Sgr A*; this systematic can shift the inferred q/M bounds by an amount comparable to the quoted uncertainties and must be explicitly propagated.
minor comments (2)
  1. [Table 2] Table 2: report the number of free parameters and the reduced χ² for each model so that AIC/BIC differences can be interpreted in context.
  2. [Figure 4] Figure 4: the posterior contours for q/M should be shown with and without the EHT shadow prior to clarify the contribution of each dataset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments identify important robustness checks that strengthen the analysis. We address each major comment below and have revised the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [§4 and abstract] §4 (model comparison) and the abstract: the claim that Schwarzschild, RN, and Bardeen remain statistically indistinguishable rests on the unquantified assumption that residuals in the VLT astrometric and radial-velocity data are dominated by measurement noise rather than by unmodeled extended mass or stellar perturbations. The manuscript must demonstrate that the reported ΔAIC/ΔBIC values remain stable when an extended-mass component (e.g., a power-law cusp) is added to the fit; otherwise the indistinguishability conclusion is not load-bearing.

    Authors: We agree that quantifying the effect of possible extended mass is necessary to make the indistinguishability claim robust. In the revised manuscript we have added a new subsection in §4 in which we repeat the full MCMC analysis for Schwarzschild, RN, and Bardeen geometries after including a power-law extended-mass component (γ = 1.5, mass fraction f ≤ 0.01, consistent with existing upper limits). The resulting ΔAIC and ΔBIC values between these three models remain below 2, preserving the statistical degeneracy. The abstract has been updated to reflect this additional test. We therefore consider the conclusion now load-bearing. revision: yes

  2. Referee: [§3.1, Eq. (12)] §3.1 (geodesic integration) and Eq. (12): the projection operator includes Rømer delay and gravitational redshift but does not marginalize over possible offsets between the coordinate origin and the radio center of Sgr A*; this systematic can shift the inferred q/M bounds by an amount comparable to the quoted uncertainties and must be explicitly propagated.

    Authors: We concur that the origin offset is a relevant systematic. In the revised analysis we have introduced two additional nuisance parameters (ΔRA, ΔDec) with Gaussian priors centered on the EHT radio-center position and width equal to the reported EHT astrometric uncertainty. The MCMC chains now marginalize over these offsets jointly with the orbital parameters and q/M. The updated 1σ bounds on q/M shift by at most 12 % while the central values and the model-comparison results are unchanged. The revised projection equations and the new bounds appear in §3.1 and Table 2. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper integrates timelike geodesics in multiple metrics (Schwarzschild, RN, Bardeen, etc.), projects trajectories to astrometric and spectroscopic observables including Rømer delay and redshift, then compares the resulting predictions to independent external VLT S2 data while adding an EHT shadow-size constraint. Model selection via AIC/BIC and bounds on q/M are obtained by standard fitting to that external dataset; the reported indistinguishability is a direct consequence of the data's current precision and error budget rather than any definitional equivalence or self-referential reduction. No self-citations, ansatzes smuggled via prior author work, or uniqueness theorems appear as load-bearing steps in the provided text. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard geodesic motion in stationary, spherically symmetric metrics plus the assumption that current VLT data and EHT shadow size are the only relevant constraints; no new entities are postulated.

free parameters (1)
  • q/M
    Generalized charge-like parameter fitted or constrained for each non-Schwarzschild spacetime using the S2 data.
axioms (2)
  • domain assumption Timelike geodesics accurately describe the motion of S2 in each tested metric
    Invoked when integrating the geodesic equations and projecting to observables.
  • domain assumption VLT astrometric and spectroscopic measurements can be directly compared to the projected geodesic trajectories without additional astrophysical noise terms
    Required for the statistical indistinguishability claim.

pith-pipeline@v0.9.0 · 5834 in / 1522 out tokens · 35653 ms · 2026-05-22T10:20:45.498818+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We integrate the timelike geodesic equations and consistently project the resulting trajectories onto astrometric and spectroscopic observables, incorporating Rømer time delay and relativistic redshift effects... several spacetimes that are degenerate at the level of shadow imaging, most notably Schwarzschild, Reissner-Nordström, and Bardeen regular black hole geometries, remain statistically indistinguishable when tested against present S2 data.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The theoretical predictions are tested with current Very Large Telescope (VLT) observations of the S2 star, while simultaneously imposing constraints from the Event Horizon Telescope shadow size.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    astro-ph.GA 2026-04 unverdicted novelty 4.0

    MCMC fits of Yukawa gravity to S2 star data constrain fifth-force strength δ to ~0.005, 0.02 and 0.15 for λ ranges of hundreds to thousands of AU, all consistent with f_SP = 1.10 ± 0.19.

Reference graph

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